A note on packing of uniform hypergraphs

TL;DR

This paper investigates the minimum total size of two uniform hypergraphs that cannot be packed into a complete hypergraph and provides a sufficient degree condition for packability.

Contribution

It introduces bounds on the minimal sum of sizes for non-packable hypergraphs and establishes a degree-based criterion for guaranteed packing.

Findings

Determined bounds for the minimal sum of sizes of non-packable hypergraphs.

Proved a degree product condition ensuring hypergraph packing.

Extended understanding from graph to hypergraph packing scenarios.

Abstract

A packing of two $k$-uniform hypergraphs $H_1$ and $H_2$ is a set $\{H_1', H_2'\}$ of edge-disjoint sub-hypergraphs of the complete $k$-uniform hypergraph $K_n^{(k)}$ such that $H_1'\cong H_1$ and $H_2'\cong H_2$. Whilst the problem of packing of graphs (i.e. 2-uniform hypergraphs) has been studied extensively since seventies with many sharp results, much less is known about packing of general hypergraphs. In this paper we attempt to find the minimum possible sum of sizes $m(n,k)$ of two $k$-uniform, $n$-vertex hypergaphs which do not pack. We also prove a sufficient condition on the product of maximum degrees, which guarantees the packing.